of commutative graph
shifts \({\bf S}_1,...,{\bf S}_d\) , i.e., \({\bf S}_k{\bf S}_{k^{\prime }}={\bf S}_{k^{\prime }}{\bf S}_k\)  for all \(1\le k,k^{\prime }\le d\) ,
where the multivariate polynomial
\(h(t_1, \ldots , t_d)=\sum _{ l_1=0}^{L_1} \cdots \sum _{ l_d=0}^{L_d} h_{l_1,\dots ,l_d} t_1^{l_1} \ldots t_d^{l_d}\) 
has polynomial coefficients \(h_{l_1,\dots ,l_d}\) , \(0\le l_k\le L_k, 1\le k\le d\) 
[1]}–[2]}, [3]}–[4]}.
On the graph \({\mathcal {G}}=(V, E)\) , a polynomial filter \({\bf A}\)  in (REF )
can be represented by a matrix \({\bf A} = (A(i,j))_{i,j\in V}\) , which
has geodesic-width no more than the degree of the polynomial \(h\) , i.e., \(\omega ({\bf A})\le \sum _{k=1}^d L_k\) .
Then we can apply the PGDA (resp. the SPGDA if \({\bf A}\)  is Hermitian) to find eigenvectors associated with any given eigenvalue (resp.
the minimal/maximal eigenvalues) on SDNs with communication range \(L\ge \sum _{k=1}^d L_k\) .
In this section, we propose iterative algorithms to determine eigenvectors associated with a polynomial graph filter, which can be implemented on an SDN with 1 as its communication range, i.e., direct communication exists between all adjacent vertices.
